Finite Group Databasebeta

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Cayley diagram

C2×#13309 #498799

order512
nilpotency class4
isoclinismisoclinic class of #13309
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#13309

Subgroups

subgroupG/H
centerC2×C4#469
commutator, Frattini, cosocleC22×C8C24
socleC22#1999
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4)(5,13)(6,17)(7,22)(8,11)(9,14)(10,20)(12,19)(15,29)(16,31)(18,28)(21,32)(23,26)(24,30)(25,33)(27,34);(3,5)(4,9)(6,18)(7,23)(8,15)(10,26)(11,24)(12,28)(13,17)(14,20)(16,19)(21,33)(22,25)(27,31)(29,32)(30,34);(9,24)(13,29)(25,28)(26,31)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72231499, #72231505, #72231524, …
maximal normal subgroup of#72231499, #72231505, #72231524, …
maximal quotient of#72252892, #72252895, #72257110, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405995);